optional, the dependent variable, required when the ODE contains derivatives of more than one unknown function

L

-

list with the coefficients of y, y', ... entering the ODE

x

-

independent variable, required only when there is more than one symbol entering the list with the ODE coefficients

Description

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The singularities command computes the regular and irregular singular points of a given homogeneous linear ODE. The ODE could be given as a standard differential equation in, say, y&ApplyFunction;x, or as a list with the coefficients of y&ApplyFunction;x&comma;y&apos;&ApplyFunction;x&comma;y''&ApplyFunction;x&comma;... (see DEtools[convertAlg]).

•

Given a nth order linear homogeneous ODE with rational coefficients Ai, i ranging from 0 to n and An&equals;1,

The regular&equals;&lcub;...&rcub; and irregular&equals;&lcub;...&rcub; equations are present in the output regardless of the sets in their right-hand sides being empty. The equation FAIL&equals;&lcub;...&rcub; is present only when the command failed in classifying some of the singular points.

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The nature of the point x0&equals;∞ is determined by changing variables x&equals;1t: the original ODE in x has a (regular or irregular) singularity at infinity whenever the changed ODE in t has a (regular or irregular) singularity at t0&equals;0.

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This function is part of the DEtools package, and so it can be used in the form singularities(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[singularities](..).

Fractional linear transformations, also called Mobius transformations, do not change the structure of the singularities, they only move the locations of the poles. So, this other equation, obtained by changing variables x -> α&InvisibleTimes;x&plus;βγ&InvisibleTimes;x&plus;δ in the Bessel_ODE, also has one regular and one irregular singularity: